Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Tyler Jarvis

October 5th, 1999

Abstract

I will discuss joint work with T. Kimura and A. Vaintrob toward the proof of the the generalized Witten conjecture associated to KdV_{r} (the rth Gelfand-Dickey hierarchy). In particular, we give a proof of the conjecture in genus zero. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_{r} equations. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology.